GELE Adjustment Computations (Least Squares) — Least Squares — Observation EquationsConcept Map

Concept

Least Squares Principle

Sub Concepts

• Minimize weighted sum of squared residuals
• Random error theory
• Maximum likelihood principle
• Normal distribution assumption

Relationship To Central

Foundation of the entire adjustment methodology; defines the objective function to minimize

Concept

Observation Equation Matrix Form

Sub Concepts

• Design matrix A (partial derivatives)
• Observation vector l (observed minus computed)
• Parameter vector x-hat (corrections to unknowns)
• Residual vector v (computed minus observed)
• Weight matrix P (diagonal, inverse variance)

Relationship To Central

The mathematical framework that structures the observation equations for matrix solution

Concept

Normal Equations and Solutions

Sub Concepts

• Normal equation matrix form
• Solution formula (ATPAx = ATPl)
• Inverse computation (ATA)^-1
• Variance-covariance matrix

Relationship To Central

Derived from the least squares principle; provides the computational framework to solve for unknowns

Concept

Redundancy and Degrees of Freedom

Sub Concepts

• Number of observations n
• Number of unknowns u
• Redundancy formula r = n - u
• Minimum requirement r >= 1

Relationship To Central

Determines whether adjustment is possible and enables quality control through variance estimation

Concept

Reference Variance and Quality Control

Sub Concepts

• Reference variance formula
• Variance of unit weight sigma-zero
• Weighted sum of squared residuals
• Goodness of fit assessment

Relationship To Central

Provides statistical validation that the adjustment is appropriate and weights are realistic

Concept

Single Unknown (Weighted Mean) Case

Sub Concepts

• One parameter adjustment
• Weighted mean formula
• Weight normalization
• Practical survey examples

Relationship To Central

Special case simplification of the general least squares solution; most common in practical surveys

Concept

Geodetic Survey Applications

Sub Concepts

• Leveling network adjustment (PRS92)
• Traverse closure adjustment
• Triangulation/trilateration network
• GNSS baseline adjustment
• Compliance with RA 4374 requirements

Relationship To Central

Real-world context where observation equations are applied in Philippine geodetic practice

Concept

Practical Board-Exam Pitfalls

Sub Concepts

• Residual sign definition error
• Weight matrix P construction
• Redundancy verification
• Matrix dimension checking

Relationship To Central

Common mistakes that must be avoided in examination and professional practice

To

Observation Equation Matrix Form

From

Least Squares Principle

Strength

strong

Relationship

The principle (minimize weighted residuals) is implemented through the matrix formulation of observation equations

To

Normal Equations and Solutions

From

Observation Equation Matrix Form

Strength

strong

Relationship

Observation equations are differentiated with respect to unknowns to derive the normal equations

To

Redundancy and Degrees of Freedom

From

Normal Equations and Solutions

Strength

strong

Relationship

Redundancy determines if the normal equations are solvable and invertible (n > u required)

To

Reference Variance and Quality Control

From

Redundancy and Degrees of Freedom

Strength

strong

Relationship

Redundancy is the denominator in the reference variance formula; more redundancy enables better quality assessment

To

Reference Variance and Quality Control

From

Normal Equations and Solutions

Strength

strong

Relationship

The residuals from the normal equation solution are used to compute the reference variance

To

Normal Equations and Solutions

From

Single Unknown (Weighted Mean) Case

Strength

strong

Relationship

The weighted mean is a special case (u=1) of the general normal equation solution

To

Redundancy and Degrees of Freedom

From

Single Unknown (Weighted Mean) Case

Strength

moderate

Relationship

For single unknown with n observations: redundancy r = n - 1; example of minimal adjustment case

To

Geodetic Survey Applications

From

Observation Equation Matrix Form

Strength

strong

Relationship

Survey applications (leveling, traverse, GNSS) provide the practical context for setting up observation equations

To

Practical Board-Exam Pitfalls

From

Geodetic Survey Applications

Strength

moderate

Relationship

Common mistakes occur when applying general theory to specific geodetic scenarios

To

Practical Board-Exam Pitfalls

From

Observation Equation Matrix Form

Strength

moderate

Relationship

Residual sign definition, weight matrix structure, and matrix dimensions are frequent sources of error

To

Geodetic Survey Applications

From

Reference Variance and Quality Control

Strength

moderate

Relationship

Quality control through reference variance is essential for validating adjustment results per RA 4374 requirements

To

Reference Variance and Quality Control

From

Least Squares Principle

Strength

moderate

Relationship

The principle assumes normal distribution; reference variance is the statistical validation of this assumption